Generalized noise cancellation in a communication channel

ABSTRACT

A generalized noise canceler for canceling noise in a channel having noise composed of correlated noise components. The noise canceler is realized with a finite number of cascade networks, each network being composed of an inner product filter is series with another filter wherein both filters have characteristics determined from a eigenvector equations expressed in terms of the channel operator and noise operator.

FIELD OF THE INVENTION

This invention relates generally to communication systems and, morespecifically, to noise cancellation in a channel wherein the correlationof two different components of the noise are utilized to reduce theexpected noise power of one of the components.

BACKGROUND OF THE INVENTION

In the book on information theory entitled "Information Theory andReliable Communication", authored by R. G. Gallager and published byJohn Wiley, 1968, Gallager shows how the capacity of a time-continuouschannel with intersymbol interference and colored noise may bedetermined. The time-continuous channel of interest is shown in FIG. 1wherein: channel 100 has impulse response h(t); the input time signal101 to channel 100 is s(t); one component of the output signal 103 ofchannel 100 is signal 102 given by s_(o) (t) (with s_(o) (t) being theconvolution of s(t) and h(t)); and the other component of output signal103 is additive noise 104 represented by n(t). Both components of outputsignal 103 are combined in summer 105. As shown in FIG. 2, whichincludes the frequency domain equivalent of FIG. 1, the first stepdisclosed by Gallager was that of filtering the channel output 103 withequalizer 201 to flatten the noise spectrum; equalizer 201 has atransfer function given by [N(ω)]^(-1/2), where N(ω) is the spectrum ofthe noise. With reference to FIG. 3, the white noise model equivalent toFIG. 2 is shown wherein: the equivalent channel 301 is the originalchannel frequency transfer function H(ω) divided by the square-root ofthe noise spectrum ([N(ω)]^(1/2)), and the inputs to summer 305 are flatnoise component 302 given by N_(o) and channel output 303. Then Gallagerdetermines the signal shapes that yield the least lost energy intransmission through the equivalent channel of FIG. 3. These optimuminput signals form an orthogonal set that is complete in a restrictedsense on the space of bounded energy signals at the channel input. Sincethe optimum input signals are the eigenfunctions of a singular valuedecomposition of the channel impulse response, the output signals arealso orthogonal. Thus, the result of Gallager offered the tractablefeature that the complex channel of FIG. 1 could be decomposed into anarray of parallel scalar channels as illustrated in FIG. 4.

In FIG. 4, a_(i) (e.g., 401,402) is the coefficient in the seriesexpansion in the input signals {θ_(i) (t)} that lose least energy ontransmission through the equivalent channel 301: ##EQU1## where {θ_(i)i(;)} are normalized to have unit energy. With this input, the channeloutput, s_(o) (;), is given by ##EQU2## where λ_(i) ^(1/2) ψ_(i) (;) isthe channel output when the channel input is θ_(i) (;). The functions{ψ_(i) (;)} are normalized to have unit energy by using thenormalization constant λ_(i), which is the channel gain (λ_(i) =energyout/energy in) when θ_(i) (;) is transmitted. A receiver matched torecover s(;) would equalize s_(o) (;) to eliminate the λ_(i) ^(1/2)factors. This would have the effect of producing noise in thecoefficients of the final output (e.g. 407,408 from summers 405 and 406,respectively) that are proportional to ##EQU3## (e.g., 403,404) whereN_(o) is the flattened noise power spectral density.

In a separate study as presented in "The optimum combination of blockcodes and receivers for arbitrary channels," authored by the presentinventor J. W. Lechleider and published in the IEEE Trans. Commun., vol.38, no. 5, May, 1990, pp. 615-621, Lechleider investigates transmissionof short sequences of amplitude modulated pulses through a dispersivechannel with colored, added noise. Lechleider found that the channelinput sequences that led to the maximum ratio of mean output signalpower to mean noise power are the solutions to a matrix eigenvalueproblem similar to the integral equation eigenvalue problem consideredby Gallager. The structure of this channel is much like a finitedimensional version of FIG. 4. Because of the ubiquity of the form ofFIG. 4, the idea of signaling so that the transmission model is a set ofparallel, uncoupled subchannels with uncorrelated subchannel noises hascome to be known as "Structured Channel Signaling," or SCS. Thus, SCSdecomposes a complex vector channel into an ordered sequence of scalarsub-channels with uncorrelated sub-channel noise scalars. Because ofthis lack of correlation, no noise cancellation techniques can be usedto further improve the signal-to-noise performance of the total channel.This places an upper bound on what can be achieved by noise cancellationtechniques in SCS.

As discussed by Widrow et al. in the paper "Adaptive noise cancelling:Principles and applications," Proc. IEEE, vol. 63, no. 12, pp 1692-1716,December, 1975, auxiliary measurements are made of noise that arecorrelated with the noise vector that is received with the signal inorder to effect noise cancellation. The correlation is used to form abest estimate of the added noise that is subtracted from the received,noise corrupted signal. But, because of the formulation of SCS, SCSobviates any putative noise-cancellation improvement.

As alluded to in the foregoing background, SCS is a modeling techniquefor dispersive channels that provides insight into channel performance.Moreover, SCS may also be used as a basis for the design ofcommunication systems. By spreading signals over time and frequency, SCSoffers some immunity to structured noise such as impulse noise andnarrow-band noise. SCS also offers selective use of the best performingsub-channels for the most important subset of information to betransmitted. SCS subsumes generalized noise cancellation, which is atechnique for exploiting the correlation of two different components ofnoise to reduce the expected noise power of one of the components.

The art is devoid of teachings or suggestions, however, of a methodologyand concomitant circuitry for generalized noise cancellation in acommunication channel having correlated noise components.

SUMMARY OF THE INVENTION

The shortcomings of the prior art with respect to generalized noisecancellation is obviated, in accordance with the present invention, by anoise canceler realized with only a finite number of optimal noisefunctionals. The canceler and concomitant methodology of generalizednoise cancellation utilize an analysis of the formulation and thesolution of the SCS regime as the point of departure.

Broadly, in one embodiment, the generalized noise canceler forprocessing a channel output signal composed of a signal plus noise toproduce a filtered channel output signal, the noise being a linearcombination of first and second noise components, includes a parallelarrangement of a plurality of cascade networks, each cascade networkbeing responsive to the signal and the noise. Further, each cascadenetwork is composed of an inner product filter circuit having a filtercharacteristic determined from a first eigenvector equationcorresponding to minimization of noise in the first noise component, inseries with a second filter having a characteristic determined from asecond eigenvector equation related to the first eigenvector equation.All outputs of the second filters are combined to produce a filteredchannel output signal.

Broadly, in a second embodiment, a generalized noise canceler forprocessing a channel output signal composed of a signal plus noise toproduce a filtered channel output signal, the noise being a linearcombination of first and second noise components, includes a parallelarrangement of a plurality of cascade circuits, each cascade networkbeing responsive to the signal and the noise. Further, each cascadenetwork is composed of a first filter having a filter characteristicdetermined from a first eigenvector equation corresponding tominimization of the noise in the first noise component, a sampler inseries with the first filter, and a second filter having acharacteristic determined from a second eigenvector equation related tothe first eigenvector equation. All outputs of the second filters arecombined to produce a filtered channel output signal.

The organization and operation of this invention will be understood froma consideration of the detailed description of the illustrativeembodiments, which follow, when taken in conjunction with theaccompanying drawing.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 depicts a prior art time-continuous channel for which optimumsignal selection is desired;

FIG. 2 depicts a filtering arrangement applied to FIG. 1 to flatten thenoise spectrum;

FIG. 3 depicts a fiat noise channel which is equivalent to the equalizedchannel of FIG. 2;

FIG. 4 depicts an equivalent to the channel of FIG. 3 which isimplemented as an array of scalar channels;

FIG. 5 is one version of a general Hilbert space linear communicationchannel;

FIG. 6 is an illustrative embodiment of a generalized Hilbert spacenoise canceler; and

FIG. 7 is an illustrative embodiment of the generalized Hilbert spacenoise canceler of FIG. 6 depicting one realization of the inner productfilter of FIG. 6.

DETAILED DESCRIPTION

The first part of this description presents a derivation of SCS fordigital signaling of one-shot channels that is more general than thosegiven previously in the prior art. The derivation is given in a Hilbertspace setting that is applicable to a wide range of applications. Then ageneralized form of optimum noise cancellation is presented for the sameHilbert space framework. Finally, the development is applied to obtain amethodology and circuitry for noise cancellation.

I.1 STRUCTURED CHANNEL SIGNALING

The model of a communications system that is used in this discussion isillustrated in FIG. 5. This model is a generalization of the timecontinuous model studied by Gallager. The data 501 to be transmitted isa single number, a, that modulates a signal S (502), which is taken tobe an element of a Hilbert space. For example, the signal space mayconsist of all functions of time that are square-integrable on a givensegment of the time line. The channel, C (503), is a map from oneHilbert space to another. The space of received signals may, in fact,have a different form for the inner product of two elements than thetransmitted signal space does. For example, the transmitted and receivedsignal spaces may have different dimensionalities. It is assumed thatthere are no input signals that have no output. This is no loss ofgenerality; such signals would not be used in practice. The noise, n(504), is an element of the output space that is selected at random.Only the second order statistics of this noise are used. There may be asubset of the receiver input space that cannot be reached by anytransmitted signal. This subspace is referred to as the null outputspace. The part of n that lies in the null output space may becorrelated with the part that lies in its orthogonal complement. Thiscorrelation is exploited by the receiver in both SCS and noisecancellation. The receiver 506 is linear, providing an estimate, a (507)of the transmitted signal. This estimate may be written in the form

    a═<R, aS.sub.o +n>                                     (3)

where the parentheses <> indicate Hilbert space inner product, R is anelement of the output signal space, as are S_(o) and n. The channeloperator may be explicitly included in equation (3) to yield

    a═(R,aCS+n)═a(R,CS)+(R,n)                          (4)

The mean-square error in this estimate is

    e═<(a-a).sup.2 >═<a.sup.2 >[(R,CS)-1].sup.2 +<(R,n).sup.2 >](5)

The received signal power is

    <a.sup.2 >═<a.sup.2 >(R, CS).sup.2                     (6)

In SCS the signal, S, and the receiver vector, R, are jointly selectedto maximize the signal-to-noise ratio (SNR) at the receiver output,i.e., ##EQU4## However, this ratio can be made arbitrarily large byusing distortionless transmission ((R,CS)-1) and arbitrarily largetransmitted power. Thus, it is necessary to constrain the transmittedpower to, say, <a² >P while determining the best combination of S and R.Thus, a modified SNR is considered, namely, λ.sub.μ given by ##EQU5##where μ is a parameter similar to a Lagrange parameter; it will bechosen to make the transmitted power equal to <a² >P so that the secondterm in the numerator on the right in equation (7) vanishes. In equation(7), parentheses have been used to indicate inner products in the inputspace.

Now, it is noted that <(R ,n)² > is a symmetric quadratic form in R and,consequently, may be written in the form (R,NR), where N is a symmetricoperator. This converts equation (7) to the form ##EQU6##

Now, consider the combination of S and R that maximizes λ.sub.μ when μis chosen so that the the input power is <a² >P. First, keep R fixed andvary S in the sense of the calculus of variations and set the resultequal to zero to get a stationary condition. This yields

    [α-λ.sub.μ (α-1)]CS-λ.sub.μ NR═0 (9)

where

    α═(R,CS)                                         (10)

is a measure of the distortion incurred in transmission. Next, hold Sfixed in equation (8) and vary R. Then, stationarity requires that

    <a.sup.2 >[α-λ.sub.μ (α-1)]CS-λ.sub.μ NR═0                                                  (11)

Equations (9) and (11) must be satisfied if R and S are optimumreceiver-signal pair. Before proceeding with the solution of theseequations, α and μ are now determined. To do this, first take an innerproduct of equation (11) with R to get, after using equation (10),##EQU7## Now, use equation (12) in equation (7) and assume that thetransmitted power constraint is satisfied. The result is

    λ.sub.μ (1-α)═0                        (13)

The only non-trivial solution of this equation is

    α═1                                              (14)

so that the optimal solution uses distortionless transmission. The valueof μ is now easily determined by using equation (14) in equation (9) andtaking the inner product of the result with S to get ##EQU8## Equations(14) and (15) permit writing equations (9) and (11) in the form

    PC.sup.T R-S═0                                         (16)

and

    <a.sup.2 >CS-λNR═0.                             (17)

Using equations (16) in (17) then yields the following eigenvalueproblem:

    P<a.sup.2 >CC.sup.T θ.sub.i ═λ.sub.i Nθ.sub.i (18)

where θ_(i) and λ_(i) are the i^(th) eigenvector-eigenvalue pair. If itis assumed that P<a² >, which is the total transmitted power, is unity,λ_(i) becomes the SNR for unit transmitted power and equation (18)becomes

    CC.sup.T θ.sub.i ═λ.sub.i Nθ.sub.i. (19)

At this point it can be assumed that N is positive definite. However,recall that there might be a subspace of the receiver input space thatis inaccessible by transmission through the channel. This implies thatC^(T) may have a null subspace so that there might be eigenvectors ofequation (19) that correspond to a zero eigenvalue. The null sub-spaceof C^(T) is, of course, spanned by these eigenvectors. The vectors{C^(T) θ_(i) } span the range of C^(T), or, equivalently, the domain ofC. Hence, write

    ψ.sub.i ═C.sup.T θ.sub.i                     (20)

so that the ψ_(i) span the input signal space. Using equation (20)inequation (19) yields

    N.sup.-1 Cψ.sub.i ═λ.sub.i θ.sub.i    (21)

or, operating on both sides with C^(T),

    C.sup.T N.sup.-1 Cψ.sub.i ═λ.sub.i ψ.sub.i. (22)

Thus, the {ψ_(i) } also satisfy an eigenvalue problem. The form ofequation (22) is the same as that of the integral equation used byGallager. Thus, Gallager's eigenfunctions maximize the SNR at thereceiver input for constrained transmitted power.

Because of the symmetry of the operators in equation (22), the {ψ_(i) }form a complete orthogonal family on the space of input signals.Equation (22) leads to:

    (ψ.sub.i, ψ.sub.j)═Pδ.sub.ij             (23)

by standard arguments, where δ_(ij) a Kronecker delta. Using thedefinition of the {ψ_(i) } that was given by equation (20) in equation(23) gives an orthogonality principle for the {θ_(i) }, but with aweighting operator:

    (θ.sub.i, CC.sup.T θ.sub.j)═Pδ.sub.ij. (24)

Using equation (19) in (24) yields another form for this principle:##EQU9## when λ_(i) ≠0. Recalling the definition of N, equation (25)implies that ##EQU10## Thus, the noise scalar at the output of receiverθ_(i) is uncorrelated with the noise at the output of receiver θ_(j). Itis important to note that equation (23) says nothing about the componentof the noise vector that lies in the orthogonal complement of the spacespanned by the {θ_(i) } with non-zero λ_(i).

An orthogonality principle for the channel outputs corresponding theoptimal channel inputs can also be derived. To do this, take the innerproduct of both sides equation (22) with ψ_(j) to get

    (Cψ.sub.j, N.sup.-1 Cψ.sub.i)═λ.sub.i Pδ.sub.ij (27)

where equation (23) has been used.

The most important orthogonality principle resulting from the eigenvalueproblem is that the optimum channel output signal set {Cψ_(i) }, isbiorthogonal in the optimum receiver vector set {θ_(i) }. This principleimplies that the separate sub-channels formed by the {θ_(i), ψ_(i) }pairs are not coupled to each other. To arrive at this bi-orthogonality,recognize that equation (24) may be written in the form

    (θ.sub.i, Cψ.sub.j)═Pδ.sub.ij          (28)

by employing equation (20). Equation (28) says that separate informationmay be sent on each of the sub-channels without any interference betweenthem. When this is combined with uncorrelated sub-channel noise, acomplex channel with colored noise can be used as separate, unrelatedsubchannels.

I.1.1 CHANNEL CAPACITY

This section uses the SCS model to determine the capacity of the channelunder the assumption of Gaussian noise. Then the individual scalarsub-channels of the SCS model have Gaussian noise. Since the capacity ofany scalar channel is bounded below by the channel that has Gaussiannoise with the same mean-square value as noise that is actually present(e.g. see the text entitled "Transmission of Information" by R. M. Fano,John Wiley, 1973), a lower bound on the capacity of the channel isobtained by making the Gaussian assumption.

It is shown that the channel capacity is achieved when only a finitenumber of the optimal transmitted {φ_(n) } are employed in modulatingdata, so that the optimal transmitted signal space isfinite-dimensional. Thus, in a very broad class of communicationschannels are, effectively, finite-dimensional channels. This simplifiesfiltering and signal processing generally. This detailed descriptionshows how to implement this optimal and simplest form of signalprocessing.

With the assumption of Gaussian noise for the sub-channel, thesub-channel capacity for the m^(th) channel may be written in the form##EQU11## where, from equation (I.1.26), the noise on the m^(th)sub-channel is

    σ.sup.2 .sub.m ═<(θ.sub.m, n).sup.2 >      (2)

and where the information power has been written in the form A² _(m) forconvenience.

Since the noise components on different sub-channels are uncorrelated,the capacity of the overall channel, C, can be written as the sum of thecapacities of the individual sub-channels: ##EQU12##

To maximize the capacity of the channel, the power on each of thesubchannels, A² _(m), must be chosen appropriately. Of course, the sumof the sub-channel powers must be constrained during the optimization.To optimize, first assume that there are only a finite number ofsub-channels that are used, say M. Then write (3) in the form ##EQU13##The values of the A² _(m) 's cannot effect the second sum on the rightin equation (4), so the A² _(m) 's should be chosen to maximize thefirst summation in order to maximize the channel capacity. By theSchwartz inequality, the first summation is maximized if all of thelogarithms are equal. This can only be achieved if

    A.sup.2.sub.m +σ.sup.2.sub.m ═k,                 (5)

where k is some constant. To determine k, sum both sides of equation (5)over m to get ##EQU14## If the transmitted power is constrained to beA², i.e., ##EQU15## in equation (6), the following obtains: ##EQU16##With this, obviously positive, value for k, equation (5) yields##EQU17## In particular, this equation must be true for m═M, so that##EQU18## From equation (I.1.26) it is known that the mean square noiseon the subchannels is inversely proportional to the eigenvalues of astrictly positive definite compact operator. These eigenvalues can bearranged in descending order in a sequence that has zero as a limitpoint even though zero is not an eigenvalue, because of the propertiesof compact operators. Thus, the sequence {σ² _(m) } is a non-decreasing,unbounded sequence. Consequently, the sequence ##EQU19## is amonotonically non-increasing unbounded sequence. As a result, for all Mexceeding some minimal value, ##EQU20## so that equation (10) dictatesthat A² _(M) should be negative for all M exceeding the minimum value,which cannot follow. Consequently, the sequence {A² _(m) } of optimaltransmitted signal powers on the SCS sub-channels must terminate at thelargest M for which ##EQU21## and the capacity of the channel is thengiven by equation (4).

I.2 NOISE CANCELLATION

This section generalizes the noise cancellation theory advanced byWidrow, et al., in the article entitled "Adaptive noise canceling:Principles and application," as published in Proc. IEEE, vol. 63, no. 12pp. 1692-1716, December, 1975, to a Hilbert space setting. The essentialidea in noise cancellation is that the space that a noise vector lies inmay be divided into two orthogonal subspaces and the correlation of thecomponents of the noise in these subspaces can be exploited to reducethe magnitude of the noise in one of the subspaces. This is tantamountto decorrelating the components of the noise in the two subspaces.

Suppose that a (random) noise vector, n, lies in a Hilbert space, H,that is comprised of two orthogonal subspaces, H₁ and H₂. The noise isthus a linear combination of components in each subspace:

    n═n.sub.1 +n.sub.2 ═P.sub.1 n+P.sub.2 n,           (1),

where P_(k) is the orthogonal projection of H onto H_(k), i.e.,

    P.sub.k H═H.sub.k,                                     (2)

and

    P.sub.j P.sub.k ═P.sub.k δ.sub.ij.               (3)

The correlation of the components of the noise in the two subspaces isexpressed by covariance operators that are defined by

    <(P.sub.k n.sub.i P.sub.j n)>═TR[<P.sub.j n(P.sub.k n).sup.T >]═TR[P.sub.j <nn.sup.T >P.sub.k ]═TRN.sub.jk     (4)

where TR indicates the trace and

    N.sub.jk ═P.sub.j NP.sub.k                             (5)

is the covariance operator for the noise components in the subspaces.Now, suppose that it is desired to minimize the noise in H by exploitingthe correlation between n₁ and n₂. To do that, form a linear combinationof n₁ and a vector (called Ln₂) in H₁ that is linearly related to n₂.Thus, form a new vector n₀ in H₁ of the form

    n.sub.0 ═n.sub.1 +Ln.sub.2                             (6)

where L is a linear operator with domain H₂ and range H₁, i. e.,

    L═P.sub.1 LP.sub.2.                                    (7)

Of course, L should be chosen to minimize the mean-square value of n₀ :

    <(n.sub.0,n.sub.0)>═TR[n.sub.0 n.sub.0.sup.T ]═TR[(n.sub.1 +Ln.sub.2)(n.sub.1 +Ln.sub.2) .sup.T ].                   (8)

By a standard variational procedure, the optimum operator, call it L₀,is given by

    L.sub.0 ═-N.sub.12 N.sub.22.sup.-1 ═-P.sub.1 N.sub.12 N.sub.22.sup.-1 P.sub.2.                                  (9)

A filter, F, that operates on n and produces the minimal noise (n₀) inH₁ is given by

    F═P.sub.1 +L.sub.0 ═P.sub.1 (I-N.sub.12 N.sub.22.sup.-1 P.sub.2). (10)

To see that n₀ and n₂ are uncorrelated, use n₀ ═Fn and then,

    <(n.sub.0, n.sub.2)>═<TR[n.sub.0.sup.T n.sub.2 ]═TR[<n.sub.2 n.sub.0.sup.T >].                                         (11)

But,

    TR[<n.sub.2 n.sub.0.sup.T >]═TR[<n.sub.2 n.sup.T (P.sub.1 -N.sub.22.sup.-1 N.sub.21)>]═TR[N.sub.21 -N.sub.21 ]═0. (12)

I.3 NOISE CANCELLATION IN STRUCTURED CHANNEL SIGNALING

This section demonstrates that SCS incorporates optimum noisecancellation as described in the preceding section. In fact, noisecancellation is a limiting case of SCS. The starting point is theobservation that the space of channel outputs is a (possibly) propersubspace of the receiver input space. First, a formula is developed forthe operator that projects the receiver input space onto the channeloutput space. To do this, first write the channel outputs in the form

    S.sub.0 ═CS                                            (1)

where S₀ is a generic channel output that is caused by input signal S.To develop the projection, first operate on both sides of (1) with C^(T)to obtain

    C.sup.T S.sub.0 ═C.sup.T CS.                           (2)

Since C does not annihilate any channel inputs, C^(T) C ispositive-definite and consequently may be inverted in equation (2) toobtain

    (C.sup.T C).sup.-1 C.sup.T S.sub.0 ═S.                 (3)

Now, operate on both sides of equation (3) with C to obtain

    C(C.sup.T C).sup.-1 C.sup.T S.sub.0 ═S.sub.0.          (4)

Since equation (4) is true for all channel outputs, the operator on theleft must be an identity operator on the space of channel outputs. Sincethe domain of C^(T) is the receiver input space, it follows that theoperator

    P.sub.1 ═C(C.sup.T C).sup.-1 C.sup.T                   (5)

projects the receiver input space onto the channel output space, whichis now designated by H₁. That P₁ is an orthogonal projection followsdirectly from the fact that it is obviously symmetric and idempotent.

Equation (5) tells how to construct the operator that projects anyreceiver input onto the space of channel outputs, which is called H₁,from a knowledge of the channel characteristics. To see what theimplications of this construction are, consider equation (I.1.19) forthe optimum receiver vectors. The receiver eigenvectors, including thosefor λ_(i) ═0, span the receiver input space, of which H₁ is a propersubspace. The eigenvectors corresponding to λ_(i) ═0 lie in theorthogonal complement to H₁, which will be called H₂. To see this,generically index with a zero all eigenvectors and eigenvaluescorresponding to a zero eigenvalue. Then, equation (I.1.19) becomes, foreigenvectors corresponding to zero eigenvalue,

    CC.sup.T θ.sub.0 ═0.                             (6)

Operate on both sides of equation (6) with C(C^(T) C)⁻² C^(T) to obtain

    P.sub.1 θ.sub.0 ═0.                              (7)

Thus, those θ_(i) that correspond to zero eigenvalues lie completely inH₂. For the non-zero λ_(i), operate on both sides of equation (I.1.19)with P₁ to obtain

    CC.sup.T θ.sub.i ═λ.sub.i P.sub.1 Nθ.sub.i. (8)

Comparison of equations (8) and (I.1.19) now yields

    P.sub.1 Nθ.sub.i ═Nθ.sub.i                 (9)

which equation implies, on comparison with (I.1.19),

    P.sub.2 Nθ.sub.i ═0,                             (10)

where another orthogonal projection has been defined by

    P.sub.2 ═I-P.sub.1,                                    (11)

where I is the identity operator on the receiver input space and P₂ isprojection onto H₂. Use equation (11) to express θ₁ in equation (10),with the results

    P.sub.2 N(P.sub.1 θ.sub.i)+P.sub.2 N(P.sub.2 θ.sub.i)═0 (12)

or,

    N.sub.21 P.sub.1 θ.sub.i ═-N.sub.22 P.sub.2 θ.sub.i, (13)

so that

    P.sub.2 θ.sub.i ═-N.sub.22.sup.-1 N.sub.21 P.sub.1 θ.sub.i. (14)

Thus, if a noise vector, n, is passed through a matched filter, θ_(i),the sampled output of the filter is

    (θ.sub.i, n)═(P.sub.1 θ.sub.i, n)+(P.sub.2 θ.sub.i, n).                                                       (15)

Employ equation (14) in the second term on the right in equation (15)with the results

    (θ.sub.i,n)+(P.sub.1 θ.sub.i -N.sub.22.sup.-1 N.sub.21 P.sub.1 θ.sub.i, n)═(F.sup.T P.sub.1 θ.sub.i,n)═(P.sub.1 θ.sub.i,Fn)═(θ.sub.i, FN)                 (16)

where F is given by equation (I.2.10) and the definition of the adjointof an operator has been used. This equation is similar to the optimumnoise filter of section (I.2), but only one of the eigenvectors isinvolved. Now, recall from sections (I.1) and (I.1.1) that the optimumlinear receiver that achieves the maximum capacity for a channel is ofthe general form ##EQU22##

That is, for any signal x in the receiver's input space, the receiver'soutput, s_(x), which lies in the channel's input space, is of thegeneral form ##EQU23## Using equation (16) in equation (18) yields##EQU24## so that the optimum SCS receiver is comprised of the optimumnoise cancellation filter, F, followed by a bank of matched filters withsampled outputs that drive filters that have responses equal to theoptimal basis vectors of the input space.

It should now be noted that as the channel operator approaches thecharacteristics of a projection operator, i.e., as C→P₁, as it mightthrough equalization, that the channel outputs approach the channelinputs from equation (I.1.20), i.e., P₁ θ_(i) →ψ_(i). Consequently, asC→P₁, equation (19) becomes ##EQU25## so that the optimum SCS receiverbecomes an optimum noise canceler. Thus, an optimum noise canceler hasbeen determined in the form of a finite bank of matched filters withsampled outputs that drive filters with responses that are the optimalSCS input basis signals.

A circuit implementation of a generalized noise canceler 600 inaccordance with equation (18) is shown in FIG. 6 wherein the signal plusnoise (s and n--reference numeral (601)) serve as the input to aparallel arrangement of M cascade networks 610, . . ., 620. Each cascadenetwork (e.g., cascade 610) is similarly arranged and includes an innerproduct filter circuit (e.g. element 611) determined in correspondenceto a given θ_(k) (e.g., θ₁) in series with a filter (e.g. element 612)having a filter characteristic determined from a corresponding ψ_(K)(e.g., ψ₁). The outputs of the cascade networks 610, . . . , 620 aresummed in summer 631 to produce output signal s_(x) plus n₀ (referencenumeral 602). The θ_(k) 's and ψ_(k) 's are determined from the solutionof the eigenvalue relation set forth in equations I1.21 and I.1.22,respectively, given the channel operator C and the noise operator N. Aninner product is defined for the given Hilbert space; generally forrealizable circuits the output of each inner product circuit (e.g.,element 611) is a scalar value generated periodically. Each scalar valueserves as the input signal to the corresponding filter having an impulseresponse given by the ψ_(k) 's.

To illustrate one embodiment of an inner product circuit depicted FIG.6, reference is now made to FIG. 7 wherein, for example, inner productcircuit 611 of FIG. 6 is composed of filter 711, having an impulseresponse given by θ₁, followed by a sampler 713 sampling every τseconds. In the specific embodiment of FIG. 7, noise canceler 700 iscomposed of a plurality of cascade networks 710, . . . , 720 for whichcascade network 710 is exemplary; network 710 includes: matched filter711 having an impulse response given by θ₁ ; sampler 713 sampling everyτ seconds to produce a sampled output value; and matched filter 712having an impulse response given by ψ₁ for processing the sampled outputvalue.

It should be noted that the number of cascade networks in either FIG. 6or FIG. 7 is finite even if the channel is capable of transmittingsignals from an infinite dimensional space. When the channel is aprojector this is an important result. It says that the optimum noisecanceler will select a finite dimensional subspace of the projectorrange that is the best subspace for the representation of information inthe given noise environment. The subspace is best in the sense that itpermits the representation of the maximal amount of information in asignal in the sense of information theory. The optimal noise cancelerthus not only cancels noise, it also chooses the optimum subspace,including the dimensionality of the subspace, for the representation ofinformation.

Although the foregoing detailed description has focused ontime-invariant channels, it is also within the contemplation of one ofordinary skill in the art that the foregoing development may be readilymodified to encompass time-varying channels. In addition, rather thanjust treating time-domain signal spaces, one of ordinary skill in theart may readily contemplate adapting the theoretical development tofurther encompass spatial signal spaces.

It is to be understood that the above-described embodiments are simplyillustrative of the principles in accordance with the present invention.Other embodiments may be readily devised by those skilled in the artwhich may embody the principles in spirit and scope. Thus, it is to befurther understood that the circuit arrangements described herein arenot limited to the specific forms shown by way of illustration, but mayassume other embodiments limited only by the scope of the appendedclaims.

What is claimed is:
 1. A noise canceler for processing a channel outputsignal composed of a signal plus noise to produce a filtered channeloutput signal, the noise being a linear combination of first and secondnoise components, the canceler comprisinga parallel arrangement of aplurality of cascade networks, each cascade network being responsive tothe signal and the noise to produce an output signal, wherein eachcascade network includes: an inner product filter circuit having afilter characteristic determined from a first eigenvector equationcorresponding to minimization of noise in the first noise component, inseries with a second filter having a characteristic determined from asecond eigenvector equation related to the first eigenvector equation,and means for combining the outputs of the second filters to produce thefiltered channel output signal.
 2. The noise canceler as recited inclaim 1 wherein the channel is expressed by a channel operator C and thenoise is expressed by a noise operator N, and wherein the firsteigenvector equation is given by N⁻¹ C ψ_(i) =λ₁ θ_(i) and the secondeigenvector equation is given by C^(T) N⁻¹ Cψ_(i) =λ₁ ψ_(i), where ψ_(i)is an eigenvector representing the transmitted signal shape, λ₁ is aneigenvalue representing the signal-to-noise ratio, and θ_(i) is a vectorset representing the filter characteristic.
 3. A noise canceler forprocessing a channel output signal composed of a signal plus noise toproduce a filtered channel output signal, the noise being a linearcombination of first and second noise components, the cancelercomprisinga parallel arrangement of a plurality of cascade networks,each cascade network being responsive to the signal and the noise andincluding: a first filter having a filter characteristic determined froma first eigenvector equation corresponding to minimization of the noisein the first noise component; a sampler responsive to the first filter;and a second filter, coupled to the sampler, the second filter having acharacteristic determined from a second eigenvector equation related tothe first eigenvector equation, and means for combining the outputs ofthe second filters to produce the filtered channel output signal.
 4. Thenoise canceler as recited in claim 3 wherein the channel is expressed bya channel operator C and the noise is expressed by a noise operator N,and wherein the first eigenvector equation is given by N⁻¹ Cψ_(i) =λ₁θ_(i) and the second eigenvector equation is given by C^(T) N⁻¹ C ψ_(i)═λ₁ ψ_(i), where ψ_(i) is an eigenvector representing the transmittedsignal shape, λ₁ is an eigenvalue representing the signal-to-noiseratio, and θ_(i) is a vector set representing the filter characteristic.5. The noise canceler recited in claim 4 wherein the first filter in thek^(th) cascade network has an impulse response given by θ_(k), and thesecond filter in the k^(th) cascade network has an impulse responsegiven by ψ_(k).
 6. A method for canceling noise in a channel outputsignal composed of a signal plus noise to produce a filtered channeloutput signal, the noise being a linear combination of first and secondnoise components, the method comprising the steps ofprocessing thesignal plus noise with a plurality of cascade networks, each cascadenetwork being responsive to the signal and the noise, wherein theprocessing in each cascade network includes the steps of: processing thesignal and the noise with an inner product filter circuit having afilter characteristic determined from a first eigenvector equationcorresponding to minimization of noise in the first noise component, andprocessing the output of the inner product filter with a second filterhaving a characteristic determined from a second eigenvector equationrelated to the first eigenvector equation, and combining the outputs ofthe second filters to produce the filtered channel output signal.
 7. Themethod as recited in claim 6 wherein the channel is expressed by achannel operator C and the noise is expressed by a noise operator N, andwherein the first eigenvector equation is given by N⁻¹ Cψ_(i) =λ₁ θ_(i)and the second eigenvector equation is given by C^(T) N⁻¹ Cψ_(i) =λ₁ψ_(i), where ψ_(i) is an eigenvector representing the transmitted signalshape, λ₁ is an eigenvalue representing the signal-to-noise ratio, andθ_(i) is a vector set representing the filter characteristic.
 8. Amethod for canceling noise in a channel output signal composed of asignal plus noise to produce a filtered channel output signal, the noisebeing a linear combination of first and second noise components, themethod comprising the steps ofprocessing the signal plus noise with aplurality of cascade networks, each cascade network being responsive tothe signal and the noise, wherein the processing in each cascade networkincludes the steps of: processing the signal and the noise with a firstfilter circuit having a filter characteristic determined from a firsteigenvector equation corresponding to minimization of noise in the firstnoise component, sampling the output of the first filter at apredetermined rate to produce a sampled value, and processing thesampled value with a second filter having a characteristic determinedfrom a second eigenvector equation related to the first eigenvectorequation, and combining the outputs of the second filters to produce thefiltered channel output signal.
 9. The method as recited in claim 8wherein the channel is expressed by a channel operator C and the noiseis expressed by a noise operator N, and wherein the first eigenvectorequation is given by N⁻¹ Cψ_(i) =λ₁ θ_(i) and the second eigenvectorequation is given by C^(T) N⁻¹ Cψ_(i) =λ_(i) ψ_(i), where ψ_(i) is aneigenvector representing the transmitted signal shape, λ₁ is aneigenvalue representing the signal-to-noise ratio, and θ_(i) is a vectorset representing the filter characteristic.
 10. The method as recited inclaim 9 wherein the first filter in the k^(th) cascade network has animpulse response given by θ_(k), and the second filter in the k^(th)cascade network has an impulse response given by ψ_(k).